Mastering Circle Calculations: Area and Circumference | 掌握圆的计算：面积与周长

📚 Mastering Circle Calculations: Area and Circumference | 掌握圆的计算：面积与周长

In Cambridge KS3 Mathematics, circle calculations appear regularly in both progress tests and checkpoint exams. This article focuses on the two key formulas: circumference and area. You will learn how to identify the radius and diameter, apply the formulas accurately, and handle common exam-style questions, including semicircles and compound shapes.

在剑桥 KS3 数学中，圆的计算经常出现在阶段测试和 checkpoint 考试中。本文重点讲解两个关键公式：周长和面积。你将学会如何识别半径和直径，准确应用公式，并应对常见考试题型，包括半圆和组合图形。

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1. The Circle and Its Key Terms | 圆及其关键术语

A circle is a set of points that are all the same distance from a fixed centre. The radius (r) is the distance from the centre to any point on the circle, while the diameter (d) is the distance across the circle through the centre. The diameter is always twice the radius, so d = 2r and r = d ÷ 2.

圆是由与固定中心距离相等的所有点组成的图形。半径 (r) 是从圆心到圆上任意一点的距离，而直径 (d) 是穿过圆心连接圆上两点的距离。直径总是半径的两倍，因此 d = 2r，r = d ÷ 2。

You must identify the radius and diameter correctly before using any formula. A common exam question gives one and asks you to find the other, so quick recall of d = 2r is essential.

在使用任何公式之前，你必须正确识别半径和直径。考试中常见题型是给出其中一个量，让你求另一个量，因此快速掌握 d = 2r 非常关键。

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2. The Number π | 圆周率 π

The number π (pi) is the ratio of the circumference of any circle to its diameter. It is approximately 3.14 or 22/7, but it is an irrational number, so its decimal form never ends or repeats. In Cambridge KS3, you usually leave answers in terms of π unless the question asks for a decimal approximation.

圆周率 π 是任意圆的周长与其直径之比。它的近似值为 3.14 或 22/7，但它是一个无理数，因此它的小数形式永远不会终止或循环。在剑桥 KS3 中，除非题目要求给出小数近似值，否则通常将答案保留为 π 的形式。

When you do approximate, use the π button on your calculator or 3.14 if instructed. Always check whether the question wants an exact answer in terms of π or a rounded decimal.

若需要近似，请使用计算器上的 π 键，或按照题目要求使用 3.14。务必检查题目是要求用 π 表示精确答案，还是要求四舍五入后的小数。

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3. Circumference Formula | 周长公式

The circumference (C) is the distance around the circle. The two equivalent forms are:

周长 (C) 是围绕圆一周的距离。两个等价公式为：

C = πd and C = 2πr

Use C = πd when you know the diameter, and C = 2πr when you know the radius. Since d = 2r, the two formulas always give the same result.

当已知直径时使用 C = πd；当已知半径时使用 C = 2πr。因为 d = 2r，两个公式总是得到相同的结果。

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4. Area Formula | 面积公式

The area (A) of a circle is the amount of space inside it. The formula is:

圆的面积 (A) 是圆内部的空间大小。公式为：

A = πr²

Note that the area formula uses the radius only, not the diameter. If you are given the diameter, first divide by 2 to find the radius, then square it and multiply by π.

请注意，面积公式只使用半径，而不使用直径。如果题目给出的是直径，请先除以 2 求出半径，然后平方并乘以 π。

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5. Choosing Radius or Diameter | 选择半径或直径

Before calculating, highlight the given measurement and label it as radius or diameter. If the question says ‘the radius is 5 cm’, use r = 5. If it says ‘the diameter is 12 cm’, use d = 12 or r = 6.

在计算之前，先标出已知数据，并注明它是半径还是直径。如果题目说“半径为 5 cm”，则使用 r = 5。如果题目说“直径为 12 cm”，则使用 d = 12 或 r = 6。

Writing down r = … and d = … at the start of your working helps you avoid the common error of using the diameter in the area formula.

在解题开始时先写出 r = … 和 d = …，可以帮助你避免在面积公式中误用直径的常见错误。

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6. Worked Example: Circumference | 周长例题

Question: A circle has radius 7 cm. Work out the circumference. Give your answer in terms of π and as a decimal to 1 decimal place.

题目：一个圆的半径为 7 cm。求它的周长。答案分别用 π 的形式和保留 1 位小数的形式表示。

Step 1: Use C = 2πr. Substitute r = 7: C = 2 × π × 7 = 14π cm. Step 2: Use π ≈ 3.14159: C = 14 × 3.14159 ≈ 43.982 cm, which is 44.0 cm to 1 d.p.

步骤 1：使用 C = 2πr。代入 r = 7：C = 2 × π × 7 = 14π cm。步骤 2：使用 π ≈ 3.14159：C = 14 × 3.14159 ≈ 43.982 cm，保留 1 位小数为 44.0 cm。

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7. Worked Example: Area | 面积例题

Question: A circle has diameter 10 cm. Work out the area. Give your answer to 1 decimal place.

题目：一个圆的直径为 10 cm。求它的面积，答案保留 1 位小数。

Step 1: Find the radius: r = d ÷ 2 = 10 ÷ 2 = 5 cm. Step 2: Use A = πr². Substitute r = 5: A = π × 5² = π × 25 = 25π cm². Step 3: 25π ≈ 25 × 3.14159 ≈ 78.54 cm², so 78.5 cm² to 1 d.p.

步骤 1：求半径：r = d ÷ 2 = 10 ÷ 2 = 5 cm。步骤 2：使用 A = πr²。代入 r = 5：A = π × 5² = π × 25 = 25π cm²。步骤 3：25π ≈ 25 × 3.14159 ≈ 78.54 cm²，保留 1 位小数为 78.5 cm²。

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8. Compound Shapes with Circles | 含圆的组合图形

Some exam questions combine circles with rectangles, triangles or semicircles. For a semicircle, find the area of the full circle and divide by 2. For the perimeter of a semicircle, add the curved part (πr) and the straight diameter (2r).

一些考试题目会把圆与矩形、三角形或半圆组合在一起。对于半圆，先求整个圆的面积再除以 2。对于半圆的周长，需要将弯曲部分 (πr) 与直边直径 (2r) 相加。

Example: A shape is made from a rectangle of length 8 cm and width 4 cm with a semicircle of diameter 4 cm added on one side. The total area is area of

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